A Comprehensive Approach for Improving the Shrinkage Estimators of a Multivariate Normal Mean Using the Particle Filter

Abstract In this article, we propose a comprehensive approach to improving shrinkage estimators of a multivariate normal mean, addressing the challenge of estimating observation error in scenarios with limited sample sizes. Our method integrates an objective Bayesian framework, particle filtering, and computational statistical techniques. First, we adopt an objective Bayesian approach by using the distribution of the observation error as a prior and generating random numbers from the standard normal distribution. This enables the definition of a density function for the squared norm of the observation vector, which follows a noncentral chi-square distribution. Next, particle filtering is employed to estimate the observation error. Finally, we construct an overall estimator that improves the James-Stein positive-part estimator (JSPPE) and the maximum likelihood estimator (MLE). Simulation results demonstrate that the proposed estimator outperforms the JSPPE in higher dimensions and the MLE in lower dimensions.